New Instanton Effects in String Theory

Chris Beasley; Edward Witten

REVIEW

New Instanton Effects in String Theory

Chris Beasley, Edward Witten

High Energy Physics - Theory · Submitted 5 Dec 2005 · 47 references

Pith's verdict: A genuine new class of instanton F-terms, with one acknowledged gap — nonvanishing of the deformation after integrating over a rational family base — that the paper does not close.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 8 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/0512039 (version 2)

Pith IDGUYK57UL, Pith's permanent record of this paper and its review

CitationChris Beasley, Edward Witten. "New Instanton Effects in String Theory." arXiv:hep-th/0512039 (2005). https://pith.science/paper/GUYK57UL

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Bottom line: the idea is real and the machinery is sound, but the step that turns a local computation into a nonzero deformation of moduli space is asserted rather than shown, and the authors say so themselves. That is the thing to press on, not the parts the reader flagged as fragile. What is actually new: a degree-one multi-fermion F-term generated by a one-parameter family of rational curves in an N=1 heterotic background, computed in physical gauge. The N=2 warm-up on E×Y is the cleanest part of the paper — there the answer is manifestly nonzero and the geometric interpretation of the tensor J acting on the partition function is convincing. The Type IIA re-derivations in Section 5 are the best evidence that the formalism is right: the (W^2)^g genus-g result and the open-string holonomy combinatorics reproducing the Dijkgraaf–Vafa expectations come out of the same machinery, and they land where they should. That is a real check, not decoration. Where it is soft: the stress-test note lands. In the E×Y example the integrand in (4.44) is constant on the elliptic fiber, so the integral over the family base collapses to Area(E)·̟ and inherits nonvanishing for free. In the N=1 case the base is rational and ̟(t,t̄) varies, so ∫_B d²t g^{tt̄} ̟ can cancel even when every local ̟ is nonzero. The paper concedes this in §4.2 — 'we have not shown directly that ω is associated to a nontrivial class in H¹_∂(M0, TM0)' — and offers the N=2 example as the reason for optimism, which is exactly the step that does not transfer. A rational-base example, or an argument that the integral is sign-definite, would settle it. On the reader's other worry: the couplings in (4.22) are checked against Grisaru–Nishino–Zanon, which is published, and the caveat about H-flux and torsion is stated where it belongs. I would not treat that as the weak link. Who should read it: anyone working on non-perturbative corrections to heterotic moduli spaces, and anyone who wants a clean worked example of physical-gauge instanton calculus. It deserves a serious referee, mainly to force the B-integral question into the open.

Referee report

Significance

If correct, the result broadens the class of non-perturbative effects available in N=1 string compactifications: instantons need not lift moduli, they can deform the moduli space, and the deformation is computable in the same worldvolume framework used for superpotentials. The paper's strengths are real and worth naming. The N=2 example of §4.1 is a controlled computation in which the relevant quantity is demonstrably nonzero, so the mechanism is established in at least one string background rather than merely postulated. The treatment of the heterotic B-field, the determinant line bundle, and the transformation of the one-loop factors on the universal family (§4.2) is careful and is what makes the fiberwise integral well defined. The distinction drawn in the introduction between F-terms in the Wilsonian action and the 1PI holomorphic-anomaly F-terms of Antoniadis et al. is a genuine conceptual clarification. The rederivations of the Type IIA closed- and open-string results in §5.2 are economical and independently useful. There are no free parameters or fitted quantities; the claims are structural and, in the N=2 case, falsifiable against the known hypermultiplet metric.

Major objections (3)

[§4.2, Eq. (4.44) and 'Brief Remarks'] The authors acknowledge that they 'have not shown directly that ω is associated to a nontrivial class in H¹_∂(M0, T M0)', and the stated reason for expecting nonvanishing is the N=2 example. That example does not transfer to the N=1 case as directly as the text implies. When X = E × Y the base of the family is the elliptic curve E, the integrand ̟ in (4.44) is constant on E, and the integral reduces to Area(E)·̟, so nonvanishing of the local quantity implies nonvanishing of ω. In the N=1 case the base B is rational and ̟(t,t̄) varies over B; the integral ∫_B d²t g^{tt̄} ̟(t,t̄) could in principle cancel even when every local ̟ is nonzero, and the paper offers no argument (sign-definiteness, a protected holomorphic limit, or an explicit rational-base example) that it does not. Since the headline claim of the abstract concerns compactifications with only N=1 supersymmetry, this is load-bearing. I would ask the authors either to supply a rational-base example or a nonvanishing argument, or to state explicitly in the abstract and introduction that N=1 nonvanishing is expected but not established, reserving the demonstrated statement for the N=2 case.
[§4.2, Eq. (4.22)] The worldvolume couplings in (4.22) are introduced as 'the natural guess, which can then be verified directly using the results in [42,43]', but the verification is not carried out in the manuscript; the only check given is that the couplings reduce to (4.19) when X = E × Y. Since the entire local and global computation of ̟ and ω rests on these three interaction terms, the reader is asked to take the central input on trust. The cited sources are published and I do not doubt that the terms are present, but the paper should either exhibit the map to the specific couplings of [42,43] or at least quote the relevant equations there, so that a reader can confirm the claim without reconstructing the superspace computation. The authors' own caveat that additional couplings appear in the presence of H-flux or torsion makes it more important to pin down what has and has not been included.
[§2, Eq. (2.10) and §4.2, Eq. (4.44)] Section 2 requires that the tensor ω representing the F-term be annihilated by ∂̄ for O_ω to be chiral, and that its class be nontrivial under the equivalence (2.10). The explicit representative constructed in (4.32) and (4.44) as J∘J applied to Ψ_C is not checked against either condition. Closure presumably follows from the supersymmetry of the construction, since J∘J of a globally defined function on M0 should be the image of a supercharge anticommutator, but the paper does not say so, and the argument is not made. A short paragraph establishing ∂̄ω = 0 for the constructed representative, or explaining why it is automatic, would close the gap between the general §2 framework and the concrete §4 computation.

Minor objections (8)

[References, [10]] The arXiv identifier for the companion paper is given as 'hep-th/0490149', which is not a well-formed number; it should presumably read hep-th/0409149. Please correct.
[§2, Eqs. (2.5)–(2.6)] The symbol ω is used both for the representative ω^i_j of H¹(M, TM) in (2.2) and for the symmetric two-tensor ω_{ij} in (2.6), and later ̟ is introduced in (4.23) for the local quantity. The relation between the two ω's, including the factor of 1/2 and the index raising in (2.6), would be clearer with one extra sentence.
[§3.1, Eq. (3.3)] The identifications S₊⊗S₋(N) = O ⊕ ∧²N* and S₊⊗S₊(N) = N* are stated without derivation. A brief indication of how the Calabi-Yau condition on X, restricted to C, produces these splittings would help readers who have not worked through the twisting.
[§3.2, Eq. (3.7)] The determinant of ∂_{O(-1)} is written without a prime while det′∂_O carries one. This is correct on CP¹ since O(-1) has no zero-modes, but a footnote saying so would prevent the reader from suspecting an inconsistency.
[§4.2, Eq. (4.28) and §5.1, Eq. (5.4)] The bundle in which ̟ takes values contains Ω¹_{U/M0} twice in each factor, and the text explains that the second factor arises from projecting out bosonic zero-modes. The notation would be easier to follow if the two occurrences were distinguished, for instance by labelling one as the zero-mode projection factor.
[§5.2, Eq. (5.17)] The index structure of the period-matrix contractions in (5.17) is difficult to parse on a first reading, and the simplification to (5.20) applies only to the special Cartan-valued components with distinct indices. A sentence stating explicitly that (5.17) is the general result and (5.20) a special case would prevent misreading.
[§4.2 and §5.2] The paper restricts throughout to single covers of C, and mentions a 'suitable prefactor for multiple-covers' only in passing in §5.2. Since the instanton sum in a heterotic background is the physically relevant object, a short remark on how multiple covers are weighted, or a citation to where this is treated, would be useful.
[§3.2 and §4.2] There are a few typographical errors, for instance 'correponding' in §3.2 and 'If general, if we consider a linear operator' in §4.2, which should read 'In general'. A careful pass would be worthwhile.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Worldsheet instantons in $N=1$ heterotic compactifications can generate multi-fermion F-terms, not only superpotentials, and a one-parameter family of rational curves deforms the complex structure of the moduli space rather than lifting it.

The reading

In an $N=1$ string compactification, instantons are usually given one job: generating a superpotential that lifts branches of the moduli space. This paper argues that they can do something subtler and, in the authors' view, more general: generate F-terms with more than two fermions, which leave the moduli space intact as a space while changing its complex structure. The case worked out in detail is heterotic compactification on a Calabi-Yau threefold $X$ with gauge bundle $V$, where a rational curve moving in a one-parameter family generates the degree-one F-term $S = S_{cl} + c + c.c.$, $c = S_{cl} + c.c.$, $c = c + c.c.$, $c = S_{cl} + c.c.$; more precisely, the correction is $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$; concretely, the added term is $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ — the added term is $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$. If the argument holds, instantons can reshape the geometry of the moduli space itself, a different physical outcome from lifting it, and one that matters for any claim that a compactification has no moduli.

What carries the argument

The computational engine is the physical (Green-Schwarz) gauge description of the worldsheet, in which spacetime supersymmetry is manifest and the counting of physical fermion zero-modes becomes a Dolbeault cohomology problem on $C$. For a curve with normal bundle $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, the right-moving fermions valued in $N$ supply $2p$ zero-modes when $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ has $p$ holomorphic sections. The one-loop worldvolume partition function $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ is a function on the moduli space; the F-term coefficient is obtained by acting on it twice with a functional differential operator $J$ built from the holomorphic three-form of $X$, the metric, and the moduli variations $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$. Anomaly cancellation of the heterotic $B$-field guarantees that the partition function transforms as a symmetric section of $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ over the universal family, which is what makes the fiberwise integral over the parameter curve $B$ well-defined and yields the tensor $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$.

Extended reading notes

Core claim

The paper's claim is that the familiar instanton-generated superpotential is only the simplest member of a family of F-terms that worldsheet instantons can produce. Counting fermion zero-modes on a holomorphic curve $C$ controls which term appears: an isolated rational curve carries exactly the two zero-modes needed for a superpotential; a rational curve moving in a one-parameter family carries two extra zero-modes and instead generates the degree-one multi-fermion F-term $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ — that is, the operator $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, which is the F-term encoding an infinitesimal deformation of the complex structure of the moduli space. Generalizing, a $p$-parameter family of rational curves generates the degree-$p$ multi-fermion F-term, and an isolated curve of genus $g$ generates a higher-derivative F-term, the heterotic analogue of the Type IIA $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ amplitudes. Along the way the authors rederive, in the physical-gauge formalism, known results about closed and open Type IIA worldsheet instantons.

Load-bearing premise

The $N=1$ computation rests entirely on the worldvolume interaction terms written in Eq. (4.22), which the authors present as a natural guess checked against earlier superspace results and which they note would require additional terms in backgrounds with $H$-flux or torsion.

Editorial extensions

If this is right

Rational curves that move in families, which cannot contribute to the superpotential, do contribute to the complex structure of the moduli space, so the set of instantons that matter physically is larger than the set usually counted.
There is a systematic dictionary between the dimension $p$ of a family of rational curves and the degree of the multi-fermion F-term it generates, giving a uniform way to organize instanton corrections by zero-mode count.
Isolated curves of genus $g$ generate higher-derivative F-terms, the heterotic counterpart of the Type IIA $S = S_{cl} + c + c.c.$, $c = c + c.c.$, $c = c + c.c.$, $c = c + c.c.$ chiral amplitudes.
The instanton-generated deformation generically mixes complex structure and complexified Kähler moduli, so the fibration structure of the perturbative moduli space need not survive non-perturbatively.
The same vanishing criterion applies throughout: a curve contributes only when the restriction of the gauge bundle $V$ to it is trivial, so a single bundle-geometry check decides which curves feed any of these F-terms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If multi-fermion F-terms are generic, then arguments that a compactification has no moduli based solely on a vanishing superpotential are incomplete: the moduli space can survive as a space while being rigidified or smoothed as a complex manifold.
Because the degree-$p$ F-terms are deformation-theoretic data of the moduli space, they suggest that higher-genus topological amplitudes should be recoverable from a purely geometric computation on the heterotic side, not only from the A-model.
The authors state they have not shown that the resulting tensor represents a nontrivial Dolbeault class; a natural test is to compute it in an orbifold limit where direct instanton calculations exist and check nonvanishing.
The zero-mode bookkeeping implies a general rule worth stating separately: any supersymmetric cycle whose deformation space is positive-dimensional generates some multi-fermion F-term, so brane instantons in Type II orientifolds should show the same pattern.

Original abstract

We describe a new class of instanton effects in string compactifications that preserve only N=1 supersymmetry in four dimensions. As is well-known, worldsheet or brane instantons in such a background can sometimes contribute to an effective superpotential for the moduli of the compactification. We generalize this phenomenon by showing that such instantons can also contribute to new multi-fermion and higher-derivative F-terms in the low-energy effective action. We consider in most detail the example of heterotic compactification on a Calabi-Yau threefold X with gauge bundle V, in which case we study worldsheet instanton effects that deform the complex structure of the moduli space associated to X and V. We also give new, slightly more economical derivations of some previous results about worldsheet instantons in Type IIA Calabi-Yau compactifications.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,403 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.12 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.81 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We describe a new class of instanton effects in string compactifications that preserve only N=1 supersymmetry in four dimensions. As is well-known, worldsheet or brane instantons in such a background can sometimes contribute to an effective superpotential for the moduli of the compactification. We generalize this phenomenon by showing that such instantons can also contribute to new multi-fermion and higher-derivative F-terms in the low-energy effective action. We consider in most detail the example of heterotic compactification on a Calabi-Yau threefold X with gauge bundle V, in which case we study worldsheet instanton effects that deform the complex structure of the moduli space associated to X and V. We also give new, slightly more economical derivations of some previous results about worldsheet instantons in Type IIA Calabi-Yau compactifications.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We describe a new class of instanton effects in string compactifications that preserve only N=1 supersymmetry in four dimensions.	contribution 0.99	0.88	0.44	asserted 0.99
s1	As is well-known, worldsheet or brane instantons in such a background can sometimes contribute to an effective superpotential for the moduli of the compactification.	background 1.00	0.04	0.04	hedged possibility 0.78
s2	We generalize this phenomenon by showing that such instantons can also contribute to new multi-fermion and higher-derivative F-terms in the low-energy effective action.	contribution 0.65	0.95	0.66	asserted 1.00
s3	We consider in most detail the example of heterotic compactification on a Calabi-Yau threefold X with gauge bundle V, in which case we study worldsheet instanton effects that deform the complex structure of the moduli space associated to X and V.	contribution 0.98	0.92	0.11	asserted 0.96
s4	We also give new, slightly more economical derivations of some previous results about worldsheet instantons in Type IIA Calabi-Yau compactifications.	contribution 0.97	0.90	0.04	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (47)

[1]

F. Denef, M. R. Douglas, B. Florea, A. Grassi, and S. Kachr u, “Fixing All Moduli in a Simple F-Theory Compactiﬁcation,” hep-th/0503124

View on Pith Pith review arXiv
[2]

E. I. Buchbinder, R. Donagi, and B. A. Ovrut, “Superpoten tials For Vector Bundle Moduli,” Nucl. Phys. B653 (2003) 400–420, hep-th/0205190

View on Pith Pith review arXiv 2003
[3]

E. I. Buchbinder, R. Donagi, and B. A. Ovrut, “Vector Bund le Moduli Superpotentials in Heterotic Superstrings and M-theory,” JHEP 0207 (2002) 066, hep-th/0206203

View on Pith Pith review arXiv 2002
[4]

E. Silverstein and E. Witten, “Criteria For Conformal In variance of (0 , 2) Models,” Nucl. Phys. B444 (1995) 161–190, hep-th/9503212

View on Pith Pith review arXiv 1995
[5]

A. Basu and S. Sethi, “World-sheet Stability of (0 , 2) Linear Sigma Models,” Phys. Rev. D 68 (2003) 025003, hep-th/0303066

View on Pith Pith review arXiv 2003
[6]

C. Beasley and E. Witten, “Residues and World-Sheet Inst antons,” JHEP 0310 (2003) 065, hep-th/0304115

View on Pith Pith review arXiv 2003
[7]

I. Aﬄeck, M. Dine and N. Seiberg, “Dynamical Supersymmet ry Breaking In Super- symmetric QCD,” Nucl. Phys. B 241 (1984) 493–534, “Dynamical Supersymmetry Breaking In Four-Dimensions and Its Phenomenological Impl ications,” Nucl. Phys. B256 (1985) 557–599

View on Pith 1984
[8]

N. Seiberg, “Exact Results on the Space of Vacua of Four-D imensional SUSY Gauge Theories,” Phys. Rev. D 49 (1994) 6857–6863, hep-th/9402044

View on Pith Pith review arXiv 1994
[9]

V. Kaplunovsky and J. Louis, “Field Dependent Gauge Coup lings in Locally Su- persymmetric Eﬀective Quantum Field Theories,” Nucl. Phys . B422 (1994) 57–124, hep-th/9402005

View on Pith Pith review arXiv 1994
[10]

C. Beasley and E. Witten, “New Instanton Eﬀects in Super symmetric QCD,” JHEP 0501 (2005) 056, hep-th/0490149

View on Pith 2005
[11]

M. Dine, N. Seiberg, X. G. Wen, and E. Witten, “Nonpertur bative Eﬀects on the String Worldsheet, I, II,” Nucl. Phys. B278 (1986) 769–789, Nucl. Phys. B289 (1987) 319–363

View on Pith 1986
[12]

E. Witten, “World-Sheet Corrections via D-Instantons ,” JHEP 0002 (2000) 030, hep-th/9907041

View on Pith Pith review arXiv 2000
[13]

M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Kodai ra-Spencer Theory of Grav- ity and Exact Results for Quantum String Amplitudes,” Commu n. Math. Phys. 165 (1994) 311–428, hep-th/9309140

View on Pith Pith review arXiv 1994
[14]

I. Antoniadis, E. Gava, K. S. Narain, and T. R. Taylor, “T opological Amplitudes in String Theory,” Nucl. Phys. B413 (1994) 162–184, hep-th/9307158

View on Pith Pith review arXiv 1994
[15]

N. Berkovits and C. Vafa, “N=4 Topological Strings,” Nu cl. Phys. B433 (1995) 123– 180, hep-th/9407190

View on Pith Pith review arXiv 1995
[16]

S. Kachru, S. Katz, A. E. Lawrence, and J. McGreevy, “Ope n String Instantons and Superpotentials,” Phys. Rev. D 62 (2000) 026001, hep-th/9912151. 47

View on Pith Pith review arXiv 2000
[17]

R. Dijkgraaf and C. Vafa, “A Perturbative Window Into No n-Perturbative Physics,” hep-th/0208048

View on Pith Pith review arXiv
[18]

H. Ooguri and C. Vafa, “The C-deformation of Gluino and N on-Planar Diagrams,” Adv. Theor. Math. Phys. 7 (2003) 53–85, hep-th/0302109

View on Pith Pith review arXiv 2003
[19]

H. Ooguri and C. Vafa, “Gravity Induced C-deformation, ” Adv. Theor. Math. Phys. 7 (2004) 405–417, hep-th/0303063

View on Pith Pith review arXiv 2004
[20]

R. Dijkgraaf, M. T. Grisaru, H. Ooguri, C. Vafa, and D. Za non, “Planar Gravitational Corrections For Supersymmetric Gauge Theories,” JHEP 0404 (2004) 028, hep-th/ 0310061

View on Pith 2004
[21]

I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, “Eﬀ ective Mu Term in Super- string Theory,” Nucl. Phys. B432 (1994) 187–204, hep-th/9405024

View on Pith Pith review arXiv 1994
[22]

I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, “To pological Amplitudes in Heterotic Superstring Theory,” Nucl. Phys. B476 (1996) 133–174, hep-th/9604077

View on Pith Pith review arXiv 1996
[23]

I. Antoniadis, K. S. Narain, and T. R. Taylor, “Open Stri ng Topological Amplitudes and Gaugino Masses,” Nucl. Phys. B729 (2005) 235–277, hep-th/0507244

View on Pith Pith review arXiv 2005
[24]

M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, “Holo morphic Anomalies in Topo- logical Field Theories,” Nucl. Phys. B405 (1993) 279–304, hep-th/9302103

View on Pith Pith review arXiv 1993
[25]

K. Becker, M. Becker, and A. Strominger, “Fivebranes, M embranes, and Non- Perturbative String Theory,” Nucl. Phys. B456 (1995) 130–152, hep-th/9507158

View on Pith Pith review arXiv 1995
[26]

J. A. Harvey and G. Moore, “Superpotentials and Membran e Instantons,” hep-th/ 9907026

View on Pith
[27]

M. Bershadsky, C. Vafa, and V. Sadov, “D-Branes and Topo logical Field Theories,” Nucl. Phys. B463 (1996) 420–434, hep-th/9511222

View on Pith Pith review arXiv 1996
[28]

M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, II , Cambridge Uni- versity Press, Cambridge, 1987

View on Pith 1987
[29]

J. Distler and B. R. Greene, “Aspects Of (2 , 0) String Compactiﬁcations,” Nucl. Phys. B304 (1988) 1–62

View on Pith 1988
[30]

P. S. Aspinwall, “Compactiﬁcation, Geometry and Duali ty: N = 2,” in String, Branes, and Gravity: TASI 99 , Ed. by J. Harvey, S. Kachru, and E. Silverstein, pp. 723–808 , World Scientiﬁc, Singapore, 1999, hep-th/0001001

View on Pith Pith review arXiv 1999
[31]

E. Witten, “Heterotic String Conformal Field Theory an d A-D-E Singularities,” JHEP 0002 (2000) 025, hep-th/9909229

View on Pith Pith review arXiv 2000
[32]

P. M. H. Wilson, “The Kahler Cone on Calabi-Yau Threefol ds,” Inv. Math. 107 (1992) 561–583

View on Pith 1992
[33]

S. Katz, D. Morrison, and M. R. Plesser, “Enhanced Gauge Symmetry in Type II String Theory”, Nucl. Phys. B477 (1996) 105–140, hep-th/9601108

View on Pith Pith review arXiv 1996
[34]

D. R. Morrison and C. Vafa, “Compactiﬁcations of F-Theo ry on Calabi–Yau Three- folds – I,” Nucl. Phys. B473 (1996) 74–92, hep-th/9602114. 48

View on Pith Pith review arXiv 1996
[35]

D. R. Morrison and C. Vafa, “Compactiﬁcations of F-Theo ry on Calabi–Yau Three- folds – II,” Nucl. Phys. B476 (1996) 437–469, hep-th/9603161

View on Pith Pith review arXiv 1996
[36]

J. Bagger and E. Witten, “Matter Couplings In N=2 Superg ravity ,” Nucl. Phys. B222 (1983) 1–10

View on Pith 1983
[37]

B. Feix, “Hyperkahler Metrics on Cotangent Bundles”, J . Reine Angew. Math. 532 (2001) 33–46

View on Pith 2001
[38]

D. Kaledin, “Hyperkahler Metrics on Total Spaces of Cot angent Bundles,” in D. Kaledin and M. Verbitsky, Hyperkahler Manifolds, Math. Phys. Series 12, Interna- tional Press, Somerville, MA, 1999

View on Pith 1999
[39]

S. J. Gates, Jr., and S. M. Kuzenko, “The CNM-Hypermulti plet Nexus,” Nucl. Phys. B543 (1999) 122–140, hep-th/9810137

View on Pith Pith review arXiv 1999
[40]

S. J. Gates, Jr., and S. M. Kuzenko, “4D, N = 2 Supersymmetric Oﬀ-Shell σ-Models on the Cotangent Bundles of K¨ ahler Manifolds,” Fortsch. Ph ys. 48 (2000) 115–118, hep-th/9903013

View on Pith Pith review arXiv 2000
[41]

R. Friedman, J. Morgan, and E. Witten, “Vector Bundles a nd F Theory,” Commun. Math. Phys. 187 (1997) 679–743, hep-th/9701162

View on Pith Pith review arXiv 1997
[42]

M. T. Grisaru and D. Zanon, “The Green-Schwarz Superstr ingσ-Model,” Nucl. Phys. B310 (1988) 57–78

View on Pith 1988
[43]

M. T. Grisaru, H. Nishino, and D. Zanon, “ β-Functions For the Green-Schwarz Su- perstring,” Nucl. Phys. B314 (1989) 363–389

View on Pith 1989
[44]

D. Marolf, L. Martucci, and P. J. Silva, “The Explicit Fo rm of the Eﬀective Action for F 1 and D-Branes,” Class. Quant. Grav. 21 (2004) S1385–S1390, hep-th/0404197

View on Pith Pith review arXiv 2004
[45]

D. Freed, “On Determinant Line Bundles,” in Mathematical Aspects of String Theory , Ed. by S.-T. Yau, pp. 189–238, Adv. Ser. Math. Phys. 1, World Scientiﬁc Publishing, Singapore, 1987

View on Pith 1987
[46]

D. Erdenebayar, “An Example of Higher Weight Superpote ntial Interaction in the Heterotic String on Orbifolds,” Int. J. Mod. Phys. A 10 (1995) 3627–3648, hep-th/ 9502046

View on Pith 1995
[47]

P. Griﬃths and J. Harris, Principles of Algebraic Geometry , John Wiley and Sons, Inc., New York, 1978. 49

View on Pith 1978

Forward citations

Cited by 19 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A New Construction of Calabi-Yau Manifolds: Generalized CICYs
hep-th 2015-07 conditional novelty 7.0 of 10

Allowing negative entries in CICY configuration matrices, with the defining relations imposed sequentially, yields a larger class of Calabi-Yau threefolds, including new Hodge number pairs and new manifolds.
Non-perturbative superpotentials across lines of marginal stability
hep-th 2007-11 accept novelty 7.0 of 10

Non-perturbative superpotentials from D-brane instantons stay continuous across lines of marginal stability because multi-instanton processes reconstruct the contribution of a single instanton after it decays.
Computation of D-brane instanton induced superpotential couplings - Majorana masses from string theory
hep-th 2007-03 conditional novelty 7.0 of 10

Euclidean D2-brane instantons in type IIA orientifolds generate perturbatively forbidden Majorana masses for right-handed neutrinos, computed explicitly in CFT and realized in a local SU(5) toy model at roughly 10^11 GeV.
Spacetime Instanton Corrections in 4D String Vacua - The Seesaw Mechanism for D-Brane Models
hep-th 2006-09 conditional novelty 7.0 of 10

Wrapped Euclidean D-branes in intersecting D6-brane models generate charged matter superpotential couplings, including right-handed neutrino Majorana masses that realize the seesaw mechanism.
Matter Swapping between Two Branes: Conditions for a Local Type IIB D3-D7-D3 Mechanism of Charged-Fermion Mixing via E3 Instantons
hep-th 2026-09 conditional novelty 6.0 of 10

A local D3+-D7-D3-/E3 Type IIB construction generates the charged-fermion mixing operator characteristic of two-sheeted braneworlds, with the mixing coefficient becoming computable from microscopic string data in a co...

Show 14 more citing papers

Non-invertible symmetries in the axiverse, and the imaginary wormholes
hep-th 2026-06 unverdicted novelty 6.0 of 10

Imaginary wormholes and the IDB imply that towers of BPS EFT instantons generate infinitely many superpotential terms that break non-invertible axion shift symmetries in N=1 axiverse models.
Testing F-theory GUTs with the Axiverse
hep-th 2026-06 unverdicted novelty 6.0 of 10

In F-theory GUTs, non-universal ALPs induced by hypercharge flux satisfy g_aγ/m_a well below the QCD axion prediction when gauge couplings unify near the string scale.
Supersymmetric Protection and the Swampland
hep-th 2020-03 conditional novelty 6.0 of 10

Exact vanishing of supersymmetric protected quantities in quantum gravity occurs only when the theory is an orbifold of a higher-supersymmetry theory, motivating a new Swampland conjecture.
Topological duality twist and brane instantons in F-theory
hep-th 2014-03 conditional novelty 6.0 of 10

A new topological duality twist constructs supersymmetric U(1) N=4 Yang-Mills on Kähler spaces with holomorphically varying coupling, incorporating duality walls and chiral defects, with application to F-theory brane ...
MSW Instantons
hep-th 2010-05 conditional novelty 6.0 of 10

M5-instantons in F-theory reduce to a computable two-dimensional CFT in the dP9 limit, and the chirality constraints on their superpotential contributions that appear in weakly coupled type IIb do not survive at finit...
Lifting D-Instanton Zero Modes by Recombination and Background Fluxes
hep-th 2007-08 conditional novelty 6.0 of 10

Instanton recombination and supersymmetric three-form flux do not, in general, lift the extra Goldstino modes of non-invariant D-brane instantons, so such instantons cannot generate superpotentials except in restricted cases.
D-brane Instantons on the T^6/Z_3 orientifold
hep-th 2007-04 conditional novelty 6.0 of 10

A detailed string-amplitude computation shows that only the U(4) and Sp(6)xU(2) gauge theories at the Z_3 orientifold singularity get ADS superpotentials from gauge instantons, while ED3 stringy instantons generate Ma...
Some Details On The Gopakumar-Vafa and Ooguri-Vafa Formulas
hep-th 2014-11 conditional novelty 5.0 of 10

The Gopakumar-Vafa and Ooguri-Vafa formulas follow from a one-loop Schwinger computation in a complex anti-selfdual graviphoton background whose extended supersymmetry fixes the magnetic moments of the BPS states.
Probing non-perturbative effects in M-theory
hep-th 2014-07 conditional novelty 5.0 of 10

The pole cancellation between worldsheet and membrane instantons, previously established in ABJM theory, is shown to extend to the Nf matrix model dual to M-theory on AdS4 x S7/Z_Nf, verified up to three worldsheet an...
Instantons and Holomorphic Couplings in Intersecting D-brane Models
hep-th 2007-05 conditional novelty 5.0 of 10

In intersecting D6-brane string models on a torus, the holomorphic gauge kinetic function receives one-loop corrections only from N=2 sectors, and matching this against the Kaplunovsky-Louis formula constrains the mod...
Non-perturbative SQCD Superpotentials from String Instantons
hep-th 2006-12 conditional novelty 5.0 of 10

A single E2-brane instanton placed on intersecting D6-branes reproduces the Affleck-Dine-Seiberg superpotential of SQCD with Nf = Nc - 1 flavours in the field theory limit.
Bell violation in the Sky
hep-th 2016-07 reject novelty 4.0 of 10

A time-dependent heavy-field mass during inflation is used to define a new curvature observable claimed to fix the inflation scale and bound the heavy-particle mass, though the Bell-violation link is not derived.
Notes on unoriented D-brane instantons
hep-th 2009-09 conditional novelty 2.0 of 10

Lecture notes survey unoriented D-brane instantons and argue that a four-hyperini coupling on T4/Z2 matches between heterotic worldsheet instantons and Type I D-string instantons.
D-brane Instantons in Type II String Theory
hep-th 2009-02 accept novelty 2.0 of 10

D-brane instantons provide a calculable, zero-mode-driven source of non-perturbative superpotential and other holomorphic couplings in N=1 Type II orientifold compactifications.

How later papers used it

47 later papers cite it

Evidence: some. 47 of the citing papers share no author with it, and 3 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 85 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 built on it.

Builds on it 2 papers

Showed that such instantons can generate, in addition to the results of this paper, multi-fermion couplings also for matter field superpotentials and under certain circumstances can also contribute to the superpotential.

Lifting D-Instanton Zero Modes by Recombination and Background Fluxes (2007) · 0.97 · paraphrased: the licence does not allow quoting
Reproduced in the present context the result associated with a deformation of the closed string moduli space obtained by this paper.

D-brane instanton effects in Type II orientifolds: local and global issues (2007) · 0.94 · paraphrased: the licence does not allow quoting

How the literature uses it

47 arXiv papers cite this in 86 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 64 sentences without a confident label

Builds on it 2 papers

We show that such instantons can generate, in addition to the results of [ref] , multi-fermion couplings also for matter field superpotentials and under certain circumstances can also contribute to the superpotential.

Lifting D-Instanton Zero Modes by Recombination and Background Fluxes · Introduction · 0.97
First let us reproduce the result associated with a deformation of the closed string moduli space obtained by Beasley and Witten in [ref] in the present context [ref] .

D-brane instanton effects in Type II orientifolds: local and global issues · Non-rigid $O(1)$ instantons: Superpotential and higher fermionic terms · 0.94

Uses its method or tool 1 paper

To compute such contributions one should perform the path-integral over all bosonic and fermionic fields [ref] and include also the contribution of the incoming M2-branes as in [ref] .

Hidden Selection Rules, M5-instantons and Fluxes in F-theory · M5-action and M2-insertions · 0.86

Cites it as background 16 papers

Until very recently it has been widely believed that among various possible D-instantons the only relevant are the one-instanton configurations [ref] .

Multi-Instanton Corrections to Superpotentials in Type II Compactifications · Introduction · 0.87
More recently, the Green-Schwarz formalism has been used to study such effects [ref] .

Black Hole Partition Function using Hybrid Formalism of Superstrings · $Z_BH$ · 0.84
Recently, these “exotic” configurations have been investigated [ref] – [ref] from various points of view.

Instanton effects in N=1 brane models and the Kahler metric of twisted matter · Instantonic brane effects · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

F. Denef, M. R. Douglas, B. Florea, A. Grassi, and S. Kachr u, “Fixing All Moduli in a Simple F-Theory Compactiﬁcation,” hep-th/0503124

View on Pith Pith review arXiv

[2] [2]

E. I. Buchbinder, R. Donagi, and B. A. Ovrut, “Superpoten tials For Vector Bundle Moduli,” Nucl. Phys. B653 (2003) 400–420, hep-th/0205190

View on Pith Pith review arXiv 2003

[3] [3]

E. I. Buchbinder, R. Donagi, and B. A. Ovrut, “Vector Bund le Moduli Superpotentials in Heterotic Superstrings and M-theory,” JHEP 0207 (2002) 066, hep-th/0206203

View on Pith Pith review arXiv 2002

[4] [4]

E. Silverstein and E. Witten, “Criteria For Conformal In variance of (0 , 2) Models,” Nucl. Phys. B444 (1995) 161–190, hep-th/9503212

View on Pith Pith review arXiv 1995

[5] [5]

A. Basu and S. Sethi, “World-sheet Stability of (0 , 2) Linear Sigma Models,” Phys. Rev. D 68 (2003) 025003, hep-th/0303066

View on Pith Pith review arXiv 2003

[6] [6]

C. Beasley and E. Witten, “Residues and World-Sheet Inst antons,” JHEP 0310 (2003) 065, hep-th/0304115

View on Pith Pith review arXiv 2003

[7] [7]

I. Aﬄeck, M. Dine and N. Seiberg, “Dynamical Supersymmet ry Breaking In Super- symmetric QCD,” Nucl. Phys. B 241 (1984) 493–534, “Dynamical Supersymmetry Breaking In Four-Dimensions and Its Phenomenological Impl ications,” Nucl. Phys. B256 (1985) 557–599

View on Pith 1984

[8] [8]

N. Seiberg, “Exact Results on the Space of Vacua of Four-D imensional SUSY Gauge Theories,” Phys. Rev. D 49 (1994) 6857–6863, hep-th/9402044

View on Pith Pith review arXiv 1994

[9] [9]

V. Kaplunovsky and J. Louis, “Field Dependent Gauge Coup lings in Locally Su- persymmetric Eﬀective Quantum Field Theories,” Nucl. Phys . B422 (1994) 57–124, hep-th/9402005

View on Pith Pith review arXiv 1994

[10] [10]

C. Beasley and E. Witten, “New Instanton Eﬀects in Super symmetric QCD,” JHEP 0501 (2005) 056, hep-th/0490149

View on Pith 2005

[11] [11]

M. Dine, N. Seiberg, X. G. Wen, and E. Witten, “Nonpertur bative Eﬀects on the String Worldsheet, I, II,” Nucl. Phys. B278 (1986) 769–789, Nucl. Phys. B289 (1987) 319–363

View on Pith 1986

[12] [12]

E. Witten, “World-Sheet Corrections via D-Instantons ,” JHEP 0002 (2000) 030, hep-th/9907041

View on Pith Pith review arXiv 2000

[13] [13]

M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Kodai ra-Spencer Theory of Grav- ity and Exact Results for Quantum String Amplitudes,” Commu n. Math. Phys. 165 (1994) 311–428, hep-th/9309140

View on Pith Pith review arXiv 1994

[14] [14]

I. Antoniadis, E. Gava, K. S. Narain, and T. R. Taylor, “T opological Amplitudes in String Theory,” Nucl. Phys. B413 (1994) 162–184, hep-th/9307158

View on Pith Pith review arXiv 1994

[15] [15]

N. Berkovits and C. Vafa, “N=4 Topological Strings,” Nu cl. Phys. B433 (1995) 123– 180, hep-th/9407190

View on Pith Pith review arXiv 1995

[16] [16]

S. Kachru, S. Katz, A. E. Lawrence, and J. McGreevy, “Ope n String Instantons and Superpotentials,” Phys. Rev. D 62 (2000) 026001, hep-th/9912151. 47

View on Pith Pith review arXiv 2000

[17] [17]

R. Dijkgraaf and C. Vafa, “A Perturbative Window Into No n-Perturbative Physics,” hep-th/0208048

View on Pith Pith review arXiv

[18] [18]

H. Ooguri and C. Vafa, “The C-deformation of Gluino and N on-Planar Diagrams,” Adv. Theor. Math. Phys. 7 (2003) 53–85, hep-th/0302109

View on Pith Pith review arXiv 2003

[19] [19]

H. Ooguri and C. Vafa, “Gravity Induced C-deformation, ” Adv. Theor. Math. Phys. 7 (2004) 405–417, hep-th/0303063

View on Pith Pith review arXiv 2004

[20] [20]

R. Dijkgraaf, M. T. Grisaru, H. Ooguri, C. Vafa, and D. Za non, “Planar Gravitational Corrections For Supersymmetric Gauge Theories,” JHEP 0404 (2004) 028, hep-th/ 0310061

View on Pith 2004

[21] [21]

I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, “Eﬀ ective Mu Term in Super- string Theory,” Nucl. Phys. B432 (1994) 187–204, hep-th/9405024

View on Pith Pith review arXiv 1994

[22] [22]

I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, “To pological Amplitudes in Heterotic Superstring Theory,” Nucl. Phys. B476 (1996) 133–174, hep-th/9604077

View on Pith Pith review arXiv 1996

[23] [23]

I. Antoniadis, K. S. Narain, and T. R. Taylor, “Open Stri ng Topological Amplitudes and Gaugino Masses,” Nucl. Phys. B729 (2005) 235–277, hep-th/0507244

View on Pith Pith review arXiv 2005

[24] [24]

M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, “Holo morphic Anomalies in Topo- logical Field Theories,” Nucl. Phys. B405 (1993) 279–304, hep-th/9302103

View on Pith Pith review arXiv 1993

[25] [25]

K. Becker, M. Becker, and A. Strominger, “Fivebranes, M embranes, and Non- Perturbative String Theory,” Nucl. Phys. B456 (1995) 130–152, hep-th/9507158

View on Pith Pith review arXiv 1995

[26] [26]

J. A. Harvey and G. Moore, “Superpotentials and Membran e Instantons,” hep-th/ 9907026

View on Pith

[27] [27]

M. Bershadsky, C. Vafa, and V. Sadov, “D-Branes and Topo logical Field Theories,” Nucl. Phys. B463 (1996) 420–434, hep-th/9511222

View on Pith Pith review arXiv 1996

[28] [28]

M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, II , Cambridge Uni- versity Press, Cambridge, 1987

View on Pith 1987

[29] [29]

J. Distler and B. R. Greene, “Aspects Of (2 , 0) String Compactiﬁcations,” Nucl. Phys. B304 (1988) 1–62

View on Pith 1988

[30] [30]

P. S. Aspinwall, “Compactiﬁcation, Geometry and Duali ty: N = 2,” in String, Branes, and Gravity: TASI 99 , Ed. by J. Harvey, S. Kachru, and E. Silverstein, pp. 723–808 , World Scientiﬁc, Singapore, 1999, hep-th/0001001

View on Pith Pith review arXiv 1999

[31] [31]

E. Witten, “Heterotic String Conformal Field Theory an d A-D-E Singularities,” JHEP 0002 (2000) 025, hep-th/9909229

View on Pith Pith review arXiv 2000

[32] [32]

P. M. H. Wilson, “The Kahler Cone on Calabi-Yau Threefol ds,” Inv. Math. 107 (1992) 561–583

View on Pith 1992

[33] [33]

S. Katz, D. Morrison, and M. R. Plesser, “Enhanced Gauge Symmetry in Type II String Theory”, Nucl. Phys. B477 (1996) 105–140, hep-th/9601108

View on Pith Pith review arXiv 1996

[34] [34]

D. R. Morrison and C. Vafa, “Compactiﬁcations of F-Theo ry on Calabi–Yau Three- folds – I,” Nucl. Phys. B473 (1996) 74–92, hep-th/9602114. 48

View on Pith Pith review arXiv 1996

[35] [35]

D. R. Morrison and C. Vafa, “Compactiﬁcations of F-Theo ry on Calabi–Yau Three- folds – II,” Nucl. Phys. B476 (1996) 437–469, hep-th/9603161

View on Pith Pith review arXiv 1996

[36] [36]

J. Bagger and E. Witten, “Matter Couplings In N=2 Superg ravity ,” Nucl. Phys. B222 (1983) 1–10

View on Pith 1983

[37] [37]

B. Feix, “Hyperkahler Metrics on Cotangent Bundles”, J . Reine Angew. Math. 532 (2001) 33–46

View on Pith 2001

[38] [38]

D. Kaledin, “Hyperkahler Metrics on Total Spaces of Cot angent Bundles,” in D. Kaledin and M. Verbitsky, Hyperkahler Manifolds, Math. Phys. Series 12, Interna- tional Press, Somerville, MA, 1999

View on Pith 1999

[39] [39]

S. J. Gates, Jr., and S. M. Kuzenko, “The CNM-Hypermulti plet Nexus,” Nucl. Phys. B543 (1999) 122–140, hep-th/9810137

View on Pith Pith review arXiv 1999

[40] [40]

S. J. Gates, Jr., and S. M. Kuzenko, “4D, N = 2 Supersymmetric Oﬀ-Shell σ-Models on the Cotangent Bundles of K¨ ahler Manifolds,” Fortsch. Ph ys. 48 (2000) 115–118, hep-th/9903013

View on Pith Pith review arXiv 2000

[41] [41]

R. Friedman, J. Morgan, and E. Witten, “Vector Bundles a nd F Theory,” Commun. Math. Phys. 187 (1997) 679–743, hep-th/9701162

View on Pith Pith review arXiv 1997

[42] [42]

M. T. Grisaru and D. Zanon, “The Green-Schwarz Superstr ingσ-Model,” Nucl. Phys. B310 (1988) 57–78

View on Pith 1988

[43] [43]

M. T. Grisaru, H. Nishino, and D. Zanon, “ β-Functions For the Green-Schwarz Su- perstring,” Nucl. Phys. B314 (1989) 363–389

View on Pith 1989

[44] [44]

D. Marolf, L. Martucci, and P. J. Silva, “The Explicit Fo rm of the Eﬀective Action for F 1 and D-Branes,” Class. Quant. Grav. 21 (2004) S1385–S1390, hep-th/0404197

View on Pith Pith review arXiv 2004

[45] [45]

D. Freed, “On Determinant Line Bundles,” in Mathematical Aspects of String Theory , Ed. by S.-T. Yau, pp. 189–238, Adv. Ser. Math. Phys. 1, World Scientiﬁc Publishing, Singapore, 1987

View on Pith 1987

[46] [46]

D. Erdenebayar, “An Example of Higher Weight Superpote ntial Interaction in the Heterotic String on Orbifolds,” Int. J. Mod. Phys. A 10 (1995) 3627–3648, hep-th/ 9502046

View on Pith 1995

[47] [47]

P. Griﬃths and J. Harris, Principles of Algebraic Geometry , John Wiley and Sons, Inc., New York, 1978. 49

View on Pith 1978